Preheat your oven to 350°F. Lightly grease a 9x13-inch baking dish with nonstick spray.

In a bowl, combine the shredded chicken with 1 cup of Monterey Jack cheese. Season with salt and pepper to your liking.

Distribute the chicken mixture evenly among the flour tortillas. Roll them up and place seam side down in the prepared baking dish.

In a saucepan over medium heat, melt the butter. Stir in the flour and cook for 1 minute until it forms a paste but is not browned.

Gradually pour in the chicken broth, whisking constantly to prevent lumps. Continue to cook until the sauce thickens slightly.

Remove the saucepan from the heat. Stir in the sour cream and diced green chiles until well combined. Avoid boiling to keep the sour cream from curdling.

Pour the creamy sauce over the rolled enchiladas in the baking dish.

Sprinkle the remaining 1 cup of Monterey Jack cheese over the top.

Bake in the preheated oven for 20-25 minutes, or until the cheese is melted and the sauce is bubbly.

For a golden top, broil for an additional 1-2 minutes, watching carefully to prevent burning.